L(s) = 1 | + (1.09 + 1.09i)2-s + (0.866 + 0.5i)3-s + 0.419i·4-s + (0.745 + 2.78i)5-s + (0.402 + 1.50i)6-s + (−1.80 − 1.93i)7-s + (1.73 − 1.73i)8-s + (0.499 + 0.866i)9-s + (−2.24 + 3.88i)10-s + (1.41 + 5.28i)11-s + (−0.209 + 0.363i)12-s + (−0.662 − 3.54i)13-s + (0.136 − 4.11i)14-s + (−0.745 + 2.78i)15-s + 4.66·16-s − 4.36·17-s + ⋯ |
L(s) = 1 | + (0.777 + 0.777i)2-s + (0.499 + 0.288i)3-s + 0.209i·4-s + (0.333 + 1.24i)5-s + (0.164 + 0.613i)6-s + (−0.683 − 0.730i)7-s + (0.614 − 0.614i)8-s + (0.166 + 0.288i)9-s + (−0.708 + 1.22i)10-s + (0.427 + 1.59i)11-s + (−0.0605 + 0.104i)12-s + (−0.183 − 0.982i)13-s + (0.0363 − 1.09i)14-s + (−0.192 + 0.718i)15-s + 1.16·16-s − 1.05·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.196 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.68057 + 1.37660i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.68057 + 1.37660i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 + (1.80 + 1.93i)T \) |
| 13 | \( 1 + (0.662 + 3.54i)T \) |
good | 2 | \( 1 + (-1.09 - 1.09i)T + 2iT^{2} \) |
| 5 | \( 1 + (-0.745 - 2.78i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-1.41 - 5.28i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + 4.36T + 17T^{2} \) |
| 19 | \( 1 + (1.39 + 0.373i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + 8.37iT - 23T^{2} \) |
| 29 | \( 1 + (-0.882 - 1.52i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.770 - 0.206i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (-3.86 + 3.86i)T - 37iT^{2} \) |
| 41 | \( 1 + (-3.88 - 1.03i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (5.58 + 3.22i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (8.52 - 2.28i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.139 - 0.241i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.16 + 5.16i)T + 59iT^{2} \) |
| 61 | \( 1 + (-4.10 + 2.37i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.87 + 0.502i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-11.3 + 3.04i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (3.72 - 13.9i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (0.431 - 0.746i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.29 + 4.29i)T - 83iT^{2} \) |
| 89 | \( 1 + (-4.21 - 4.21i)T + 89iT^{2} \) |
| 97 | \( 1 + (0.575 + 2.14i)T + (-84.0 + 48.5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.63473581860937197337171462939, −10.77164042085305331382070753056, −10.27119307637134884408170851494, −9.506452522168898430235564270646, −7.82614088600813406601992325535, −6.71952112412161142084288964406, −6.57627122229354777980210017149, −4.84283048253555496601507014373, −3.90793054217630226161380902575, −2.51428354276924780275523039501,
1.68955445586921780468651965367, 3.04863803096614503935002553155, 4.17165030912158020399654363533, 5.38525567591627932796271852344, 6.45984351584906432870077226363, 8.156712481477833092494655654220, 8.896940528703479994633829473280, 9.583526756839088488832771517789, 11.27073096270851684074667090810, 11.82301763860584138374723146322